Including Gravitational Stresses in Finite Element Simulations

Charles Williams, GNS Science New Zealand

CDM Workshop, 23-27 June, Stanford University

When do we need to include gravitational stresses?

•  Pressure/stress-dependent rheology. – Pressure-dependent bulk rheology (e.g.,

plasticity). – Stress-dependent fault rheology (e.g.,

friction). •  Viscoelastic simulations where we care

about vertical deformation. •  Other simulations where we care about

the absolute stress state. CDM Workshop, 23-27 June, Stanford University

Obvious (and wrong) approach to including gravitational stresses

CDM Workshop, 23-27 June, Stanford University examples/3d/hex8/step15.cfg

Turn on gravity

Obvious approach (cont.)

CDM Workshop, 23-27 June, Stanford University

σ xx =σ yy =ν1−ν

σ zz

examples/3d/hex8/step15.cfg

What is a realistic initial stress state?

CDM Workshop, 23-27 June, Stanford University

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 93, NO. Bll, PAGES 13,609-13,617, NOVEMBER 10, 1988

On the State of Lithospheric Stress in the Absence of Applied Tectonic Forces

A. MCGARR

U.S. Geological Survey, Menlo Park, California

Numerous published analyses of the nontectonic state of stress are based on Hooke's law and the boundary condition of zero horizontal deformation. This approach has been used to determine the gravitational stress state as well as the effects of processes such as erosion and temperature changes on the state of lithospheric stress. The major disadvantage of these analyses involves the assumption of lateral constraint which seems unrealistic in view of the observational fact that the crust can deform horizontally in response to applied loads. If the same problems are adclressed by assuming that the remote stress state is constant, instead of the condition of zero horizontal deformation, then the resulting stress states are entirely different and in good accord with observations. In the absence of applied tectonic forces the only likely gravitational stress states are those for which all three principal stresses are nearly equal. To the contrary, the gravitational stress states developed on the basis of the lateral constraint assumption can be ruled out. The processes of erosion and sedimentation have slight tendencies to increase and decrease, respectively, the state of deviatoric stress. In particular, for initial stress states in the range of slightly extensional to compressional, erosion has the effect of enhancing the ratio of average horizontal to vertical stress, which may explain, at least in part, the conunon observation of high near-surface horizontal stresses. Temperatm'e changes have only minor effects on the stress state, as averaged over the thickness of the lithosphere.

INTRODUCTION O'h -- O'zz

Measurements of the stress field in the crust can provide useful information about the forces responsible for various tectonic processes such as interplate motion and earthquakes [e.g., MeGart and Gay, 1978]. The interpretation of stress data is impeded, however, by the absence of a generally agreed-upon "reference state of crustal stress." That is, to determine what the stress measurements mean in terms of applied tectonic forces, it is obviously necessary to know what the state of stress would have been in the absence of such forces.

To establish a reference stress state, it is necessary to determine the most reasonable initial state of stress and

then to calculate the stress changes due to processes such as erosion, sedimentation, and temperature change. As will be seen, the resulting state of stress is one for which all three principal stresses are nearly equal. Before the presentation of this argument, however, a brief review of this problem area is in order.

Surprisingly, even today there is considerable disagree- ment about the state of stress in the absence of applied tectonic loads. Consider a flat geometry with the z coor- dinate representing depth. The vertical stress o.zz, about which there is little argument, is given by

where p is the average density between the surface and the depth z, and g is gravity. Equation (1) has a sound theoretical and observational basis [e.g., McGarr and Gay, 1978]. The disagreement involves the level of horizontal principal stresses, as represented by o.h. For the txvo most commonly assumed stress states,

This paper is not subject to U.S. copyright. Published in 1988 by the American Geophysical Union.

Paper number 88JB03388.

or

(2)

-- o.zz (3) o.n 1- v

where v is Poisson's ratio. Because y is typically close to 0.25 for crustal rocks, there is quite a substantial difference between the horizontal stress states of (2) and (3), with o.h being about 1/3 of o.z in the latter case.

A more general version of (2) is

o.1 = o'2 = o.3 -- pgz (4) where o.1, o.2, and o.3 represent the three principal stresses. This ]ithostatic stress state was favored by Anderson [1951], who opined that over long periods of time the crust deforms inelastically in response to any deviatoric stress and thus approaches the state for which there is no deviatoric stress. One of the purposes of this article is to demonstrate that (2) and (4) are the most likely nontectonic stress state, even in the absence of the long- term deformation mechanisms assumed by Anderson. First, ]towever, I describe some problems with the gravitationM stress state of (3).

The assumption of lateral, or bilateral, constraint leads to (3). As explained, for example, by Price [1966], fi'om Hooke's law

1

= -. + -. + (zz • •

where E is Young's modulus. If the horizontal principM strMns exx and eyy are set equal to zero, then the horizontM stresses, axx and ayy, are given by (3). In the words of Price

13,609

Since gravity is not ‘turned on’, and since materials are either emplaced or relaxed by inelastic processes, an isotropic initial stress state is more realistic.

σ xx =σ yy =σ zz = ρgh

Using initial stresses to provide realistic stresses/displacements

CDM Workshop, 23-27 June, Stanford University examples/3d/hex8/step16.cfg

Using initial stresses (cont.)

CDM Workshop, 23-27 June, Stanford University

σ xx =σ yy =σ zz = ρgh

examples/3d/hex8/step16.cfg

How to handle topography?

CDM Workshop, 23-27 June, Stanford University From Currenti and Williams (2014).

Uniaxial strain

Isotropic stress

Gravitational stresses with topography

① Compute an initial set of stresses by just ‘turning on’ gravity. a)  Use ν = 0.5 to represent incompressible

material (not yet possible in PyLith). b)  Apply far-field traction/displacement BC to

counteract uniaxial strain effects. ② Use stresses computed in step 1 as

initial stresses.

CDM Workshop, 23-27 June, Stanford University

Viscoelastic problems with vertical deformation

•  Will deformation be driven by gravity or by applied tractions/displacements/slip? – Do we want the initial stress state to induce

deformation? •  The default infinitesimal strain

formulation does not recompute body forces consistent with the deformed configuration.

CDM Workshop, 23-27 June, Stanford University

Simple set of ‘floating box’ examples (fun with gravity)

① Compute exact gravitational stresses and apply these as initial stresses for the viscoelastic problem.

② Assume simple initial stresses consistent with uniform density. a)  Perform simulation using default

infinitesimal strain formulation. b)  Perform simulation using finite strain

formulation. c)  Same as b) with smaller time step.

CDM Workshop, 23-27 June, Stanford University examples/2d/gravity

Problem setup

CDM Workshop, 23-27 June, Stanford University

Viscoelastic, ρv

Elastic, ρe

examples/2d/gravity

Case 1: Initial stress = exact gravitational stress

CDM Workshop, 23-27 June, Stanford University No displacement!

examples/2d/gravity/grav_stress_finite_is1.cfg

Case 2a: σxx = σyy = ρvgh Infinitesimal strain

CDM Workshop, 23-27 June, Stanford University Too much displacement!

examples/2d/gravity/grav_stress_infin_is2.cfg

Case 2b: σxx = σyy = ρvgh Finite strain

CDM Workshop, 23-27 June, Stanford University What’s going on?

examples/2d/gravity/grav_stress_finite_is2.cfg

Case 2b: σxx = σyy = ρvgh Finite strain (cont.)

CDM Workshop, 23-27 June, Stanford University ‘Drunken sailor’ instability!

Problems with finite strain solution and standard time step size

•  When we have gravity + viscoelasticity + free surface there is a known instability in the solution (‘drunken sailor’ or ‘sloshing’ instability).

•  To overcome this instability, a Courant condition (Kaus et al., 2010) prescribes a time step size much smaller than the ‘standard’ stable time step size.

CDM Workshop, 23-27 June, Stanford University

Case 2b: σxx = σyy = ρvgh Finite strain + small dt

CDM Workshop, 23-27 June, Stanford University Much smaller displacements!

examples/2d/gravity/grav_stress_finite_is3.cfg

Case 2b: σxx = σyy = ρvgh Finite strain + small dt (cont.)

CDM Workshop, 23-27 June, Stanford University Stable solution with smaller dt!

examples/2d/gravity/grav_stress_finite_is3.cfg

Summary

•  Problems with no topography and simple density distributions: – Compensate gravitational stresses with a

simple depth-varying initial stress state. •  Problems with topography and/or

complex density distributions: – Run an initial simulation to compute

equilibrium stresses and then use these as initial stresses in the actual simulation.

CDM Workshop, 23-27 June, Stanford University

Summary (cont.) •  Problems with time-varying vertical

displacement: –  If we want subsequent deformation to be

driven by a non-gravitational source: •  Apply initial stresses equivalent to the

gravitational stress. –  If we want gravity to drive the deformation:

•  Apply initial stresses that do not equilibrate with density contrasts.

– Need to use finite strain formulation, but this generally requires small time steps.

CDM Workshop, 23-27 June, Stanford University

PyLith examples of interest •  examples/3d/hex8/step15.cfg

– Gravity with no initial stresses. •  examples/3d/hex8/step16.cfg

– Gravity with isotropic initial stresses. •  examples/3d/hex8/step17.cfg

– Gravity with no initial stresses and finite strain.

•  examples/2d/gravity – 2D gravity examples shown previously.

CDM Workshop, 23-27 June, Stanford University